Solution of a truss topology bilevel programming problem by means of an inexact restoration method

We formulate a truss topology optimization problem as a bilevel programming problem and solve it by means of a line search type inexact restoration algorithm. We discuss details of the implementation and show results of numerical experiments.We formulate a truss topology optimization problem as a bilevel programming problem and solve it by means of a line search type inexact restoration algorithm. We discuss details of the implementation and show results of numerical experiments.301109125CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPERJ - FUNDAÇÃO CARLOS CHAGAS FILHO DE AMPARO À PESQUISA DO ESTADO DO RIO DE JANEIROFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOE-26/171.164/2003-APQ106/53768-

Solution Of The Urban Traffic Problem With Fixed Demand Using Inexact Restoration

Congested traffic has become a part of the day-to-day for the residents of big metropolitan centers. From an economic viewpoint, this problem has been causing huge financial damage and strategic measures must be taken to tackle it. An alternative means of solving the problem is the inclusion of toll charges on routes with a view to decongesting the road network. The mathematical formulation of this alternative involves the solving of an optimization problem with equilibrium constraints (MPEC). This work proposes an algorithm for the solution of this problem based on the strategy of inexact restoration.837-4019071918Andreani, R., Castro, S.L.C., Chela, J.L., Friedlander, A., Santos, S.A., Aninexact-restoration method for nonlinear bilevel programming problems (2009) Comput. Optim. Appl., 43, pp. 307-328Andreani, R., Martinez, J.M., Svaiter, B.F., On the Regularization of mixed complementarity problems (2000) Numerical Functional Analysis and Optimization, 21, pp. 589-600Andreani, R., Martinez, J.M., On the reformulation od Nonlinear Complementarity Problems using the Fischer-Burmeister function (1999) Applied Mathematics Letters, 12, pp. 7-12Andreani, R., Friedlander, A., Bound Constrained Smooth Optimization for Solving Variational Inequalities and Related Problems (2002) Annals of Operations Research, 116, pp. 179-198Arnott, R., Small, K., (1994) The economics of traffic congestion, , Boston College Working Papers in Economics 256, Boston College, Department of EconomicsBazarra, M.S., Sherali, H.D., Shetty, C.M., (1993) Nonlinear Programming: Theory and Algoritms, , Second Edition, John Wiley & Sons, New YorkBonnans, J.F., Shapiro, A., (2000) Perturbation Analysis of Optimization Problems, , Springer Series in Operations Research, SpringerBrotcorne, L., Labbé, M., Marcotte, P., Savard, G., A Bilevel Model for Toll Optimization on a Multicommodity Transportation Network (2001) Transportation Science, 35 (4), pp. 345-358Calamai, P.H., Vicente, L.N., Generating quadratic bilevel programming test problems (1994) ACM Transactions on Mathematical Software, 20, pp. 103-119Chela, J.L., (2006) Resolução do problem a de programao matemática com restrições de equilíbrio usando restauração inexata, , PhD thesis, University of CampinasFerrari, P., Road network toll pricing and social welfare (2002) Trans. Res. B, 36, pp. 471-483Harker, P.T., Pang, J.S., Existence of optimal solutions to mathematical programs with equilibrium constraints (1988) Operations Research Letters, 7 (2), pp. 61-64Hearn, D.W., (1980) Bounding Flows in Traffic Assignment Models, , Research report N.80-4, Dept. of Industrial and Systems Enginnering, University of Florida, Gainesville, FL 32611Hearn, D.W., Ramana, M.V., Solving congestion toll princing models (1998) Equilibrium and Advanced Transportation Modelling, pp. 109-124. , P. Marcotte, S. Nguyen (eds), Kluwer Academic Publisher, Boston, The NetherlandsHearn, D.W., Yildirim, M.B., A toll pricing framework for traffic assignment problems with elastic demands (2001) Current Trends in Transportation and Network Analysis: Miscellanea in Honor of Michael Florian, , M. Gendreau, P. Marcotte(eds), Kluwer Academic Publisher, Dordrecht, The NetherlandsHearn, D.W., Lawphongpanich, S., An MPEC approach to second-best toll pricing (2004) Mathematical Programming Series B, 101, pp. 33-55Hearn, D.W., Bergendorff, P., Ramana, M.V., Congestion Toll Pricing of Traffic Networks, Network Optimization (1997) Lecture Notes in Economics and Mathematical Systems, 450, pp. 51-71. , P. M. Pardalos, D.W. Hearn and W.W. Hager (Eds.), Springer-VerlagJohansson-Stenman, O., Sterner, T., What is the scope for environmental road pricing? (1998) Road pricing Traffic Congestion and Environment, , K.J. Button, E.T. Verhoef (eds.), Edward Elgar Publishing Limited, London, EnglandLabbé, M., Marcotte, P., Savard, G., A bilevel model of taxation and its application to optimal highway pricing (1998) Manage. Sci, 44 (12), pp. 1608-1622Migdalas, A., Bilevel Programming in traffic planning: models, methods and challenge (1994) Journal of Global Optimization, 4, pp. 340-357Patriksson, M., Rockafellar, R.T., A Mathematical model and descent algorithm for bilevel traffic management (2002) Trans. Sci, 36, pp. 271-291Solodov, M.V., Svaiter, B.F., A New Projection Method for Variational Inequality Problems (1999) SIAM Journal Control Optimization, 37, pp. 765-77

Economic inexact restoration for derivative-free expensive function minimization and applications

The Inexact Restoration approach has proved to be an adequate tool for handling the problem of minimizing an expensive function within an arbitrary feasible set by using different degrees of precision in the objective function. The Inexact Restoration framework allows one to obtain suitable convergence and complexity results for an approach that rationally combines low- and high-precision evaluations. In the present research, it is recognized that many problems with expensive objective functions are nonsmooth and, sometimes, even discontinuous. Having this in mind, the Inexact Restoration approach is extended to the nonsmooth or discontinuous case. Although optimization phases that rely on smoothness cannot be used in this case, basic convergence and complexity results are recovered. A derivative-free optimization phase is defined and the subproblems that arise at this phase are solved using a regularization approach that take advantage of different notions of stationarity. The new methodology is applied to the problem of reproducing a controlled experiment that mimics the failure of a dam

Assessing the reliability of general-purpose Inexact Restoration methods

Inexact Restoration methods have been proved to be effective to solve constrained optimization problems in which some structure of the feasible set induces a natural way of recovering feasibility from arbitrary infeasible points. Sometimes natural ways of dealing with minimization over tangent approximations of the feasible set are also employed. A recent paper Banihashemi and Kaya (2013)] suggests that the Inexact Restoration approach can be competitive with well-established nonlinear programming solvers when applied to certain control problems without any problem-oriented procedure for restoring feasibility. This result motivated us to revisit the idea of designing general-purpose Inexact Restoration methods, especially for large-scale problems. In this paper we introduce affordable algorithms of Inexact Restoration type for solving arbitrary nonlinear programming problems and we perform the first experiments that aim to assess their reliability. Initially, we define a purely local Inexact Restoration algorithm with quadratic convergence. Then, we modify the local algorithm in order to increase the chances of success of both the restoration and the optimization phase. This hybrid algorithm is intermediate between the local algorithm and a globally convergent one for which, under suitable assumptions, convergence to KKT points can be proved28

Reformulation of a nonlinear programming problem with multiobjective constraints

Orientadores: Roberto Andreani, Santos, Sandra AugustaTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Nesta tese apresentamos estratégias para resolver uma classe importante de problemas de tomada de decisões. Nosso propósito consiste em minimizar uma determinada função objetivo escalar F(x), com conjunto viável um conjunto propriamente eficiente do problema multiobjetivo min f(x), com f(x) função vetorial. Este problema é formulado como um problema em dois níveis, onde com um escalamento conveniente, podemos garantir pontos propriamente eficientes do problema vetorial do níivel inferior, usando resultados de Geoffrion (1968). Este tipo particular de ponto é útil para a tomada de decisões ótimas sem negligenciar nenhum dos objetivos da função vetorial. Também analisamos uma dificuldade presente nos métodos de Restauração Inexata na resolução de problemas com restrições de complementaridade e, mostramos o artifício para contornar a complicação. Para resolver o problema em dois níveis apresentamos estratégias de Restauração Inexata (RI) e as comparamos com a estratégia da reformulação, na qual se utilizam as condições de Karush-Kuhn-Tucker (KKT) do problema do segundo nível como restrições de um problema de programação não linear. Também apresentamos resultados numéricos que evidenciam que as estratégias de RI são melhores que a reformulação KKTAbstract: In this thesis, we present strategies for solving an important class of problems in the decision making. The problem consists in minimising a determinate objective function scalar F(x), where x is a properly efficient point of the multiobjective problem min f(x), with f(x) vectorial function. This problem is formulated as a bilevel problem, where with a convenient scaling, we can guarantee points properly efficient of the vectorial problem of the inferior level, cf. Geoffrion (1968). This particular type of point is useful for making optimum decisions without neglecting any of the objectives of the vectorial function. We have also detected a difficulty of the Inexact Restauration methods in the resolution of problems with complementarity constraints and show an artifice for solving the complication. To solve the bilevel problem, we present strategies of Inexact Restoration and compare them with the traditional strategy that uses the KKT conditions of the problem from the second level. We also present numerical experiments that evidence that the RI strategies are better than the traditional methodDoutoradoMatematica AplicadaDoutora em Matemática AplicadaCNP

An inexact-restoration method for nonlinear bilevel programming problems

Bilevel programming, Inexact-restoration, Optimization,